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Properties

Label 2-168-8.5-c3-0-1
Degree $2$
Conductor $168$
Sign $-0.626 + 0.779i$
Analytic cond. $9.91232$
Root an. cond. $3.14838$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.92 + 2.07i)2-s + 3i·3-s + (−0.577 − 7.97i)4-s + 14.7i·5-s + (−6.21 − 5.77i)6-s − 7·7-s + (17.6 + 14.1i)8-s − 9·9-s + (−30.5 − 28.3i)10-s + 38.2i·11-s + (23.9 − 1.73i)12-s − 10.4i·13-s + (13.4 − 14.4i)14-s − 44.2·15-s + (−63.3 + 9.21i)16-s − 51.5·17-s + ⋯
L(s)  = 1  + (−0.681 + 0.732i)2-s + 0.577i·3-s + (−0.0721 − 0.997i)4-s + 1.31i·5-s + (−0.422 − 0.393i)6-s − 0.377·7-s + (0.779 + 0.626i)8-s − 0.333·9-s + (−0.965 − 0.897i)10-s + 1.04i·11-s + (0.575 − 0.0416i)12-s − 0.222i·13-s + (0.257 − 0.276i)14-s − 0.761·15-s + (−0.989 + 0.143i)16-s − 0.734·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.626 + 0.779i$
Analytic conductor: \(9.91232\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :3/2),\ -0.626 + 0.779i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.229392 - 0.478687i\)
\(L(\frac12)\) \(\approx\) \(0.229392 - 0.478687i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.92 - 2.07i)T \)
3 \( 1 - 3iT \)
7 \( 1 + 7T \)
good5 \( 1 - 14.7iT - 125T^{2} \)
11 \( 1 - 38.2iT - 1.33e3T^{2} \)
13 \( 1 + 10.4iT - 2.19e3T^{2} \)
17 \( 1 + 51.5T + 4.91e3T^{2} \)
19 \( 1 - 4.22iT - 6.85e3T^{2} \)
23 \( 1 + 8.91T + 1.21e4T^{2} \)
29 \( 1 + 248. iT - 2.43e4T^{2} \)
31 \( 1 + 197.T + 2.97e4T^{2} \)
37 \( 1 + 84.7iT - 5.06e4T^{2} \)
41 \( 1 - 272.T + 6.89e4T^{2} \)
43 \( 1 + 269. iT - 7.95e4T^{2} \)
47 \( 1 + 547.T + 1.03e5T^{2} \)
53 \( 1 - 203. iT - 1.48e5T^{2} \)
59 \( 1 - 584. iT - 2.05e5T^{2} \)
61 \( 1 - 840. iT - 2.26e5T^{2} \)
67 \( 1 - 611. iT - 3.00e5T^{2} \)
71 \( 1 + 265.T + 3.57e5T^{2} \)
73 \( 1 + 158.T + 3.89e5T^{2} \)
79 \( 1 + 523.T + 4.93e5T^{2} \)
83 \( 1 - 1.29e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.46e3T + 7.04e5T^{2} \)
97 \( 1 + 157.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.21386870398376252812379767339, −11.57758777588450759961877437131, −10.59590464106126411882504568051, −9.978060711747664724014591045264, −9.042257592987935312096593373029, −7.62218455121241270244373060855, −6.81327754793074122852028361918, −5.76387714681660250626624881459, −4.21181005816229782785510852629, −2.41941909671912799677555254225, 0.30383524737494982420566318249, 1.61676520525144547269637455877, 3.35285118337076047888525906092, 4.91246325852321983555268882955, 6.50442370717775166808088002979, 7.890422490843195262649838539996, 8.777710434483038312516353974909, 9.374673493018022340576387223119, 10.85521271616265195884525391585, 11.66832995195467373825885793441

Graph of the $Z$-function along the critical line